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For medium and large gas turbine engines, the lean combustion technology has been recognised asone of the effective solutions for emission reduction. However, such systems are often susceptible to thermo-acoustic instability. As an essential component of the combustion system, the fuel injector plays a critical role in the feedback loop that leads to this instability. This thesis presents a study on the unsteady aerodynamic response of generic lean burn injector passages to incident acoustic waves. Single and two passage injector configurations were considered which consist of many of the representative features comprising a modern lean burn fuel injector. In this research a combination of experimental, analytical and numerical approaches were applied. ACFD methodology based on URANS simulations was developed by incorporating different acoustic boundary conditions in an open-source solver (OpenFOAM). In addition, a von Neumann stability type analysis was performed which provides guidelines for the mesh and numerical setups of the simulations. To assess the validity of the CFD method, simulations were performed for selected orifice geometries and a Helmholtz resonator. Good agreement between the simulation results and the available experimental data were achieved. With its accuracy verified, this numerical methodology allows the characteristics of an acoustic element to be predicted and the unsteady flow field to be examined. The overall acoustic response of the injector is characterised with the (specific) impedance. It was at first derived from experiments using the multi-microphone technique. Subsequently the experimental scenarios were reproduced in a series of CFD simulations applying the developed numerical methodology. Fidelity of the CFD simulations was confirmed by the excellent agreement between the (specific) impedance measured in experiments and predicted by the simulations respectively. Furthermore, considering the moderate computational cost of the simulations, the numerical approach developed also provides a practical design tool from the acoustic perspective for complex geometries such as gas turbine injectors. Encouraged by the accuracy in the CFD prediction of the injector impedance, the acoustically excited unsteady flow both within and downstream of the injector passages was investigated from the simulation results. This was mainly based on the temporal Fourier analysis of the unsteady flow field, from which the correlation between the flow fluctuations at different spatial locations of the flow field was extracted. In complement to the numerical investigation, an analytical study was carried out to examine the influences of certain geometric parameters of the injector passage on its acoustic characteristics. Following the results from the analysis of the baseline injector configurations, various design modifications were developed which enable the acoustic characteristics of the injector to be manipulated. These include the configurations that demonstrate circumferentially non-coherent response. The design and evaluation of the modified injectors were assisted by the numerical methodology established in this thesis. Motivated by previous experimental works on the unsteady spray responses of lean burn injectors, the current CFD method was extended to perform simulations of the two-phase flows of the two-passage baseline injector configuration, which include particles with representative properties of the fuel droplets observed in the experiments. In this investigation, the droplet sizes were assumed to be fixed and only one-way coupling from the continuous phase to the discrete phase is enabled. Two injections were considered, one with a uniform droplet size and the other with a Rosin-Rammler size distribution. The statistics of the droplets at different planes downstream of the injector was analysed and showed qualitative agreements with previous experimental results.